### Principle of Active filtering

The block diagram of a BMT with CANE is shown in Fig. 3. The original complex baseband signal is modulated by an EDSM modulator to a bitstream envelope fashion, and then, both the I and Q paths are split into multiple channels with different delays. These delayed signals are then up-converted with the identical local oscillator (LO), amplified and combined.

Let the normalized equivalent baseband filter impulse response be denoted by \({h}_{B}(t)\) and the corresponding filter transfer function by \({H}_{B}(f)\). To simplify the analysis, the combining coefficients \({C}_{k}\) are assumed to be positive real numbers. Based on the block diagram in Fig. 3, \({h}_{B}(t)\) and \({H}_{RF}(f)\) are given by

$$\begin{array}{*{20}c} {h_{B} \left( t \right) = \mathop \sum \limits_{k = 0}^{N - 1} C_{k} \delta \left( {t - k\tau } \right)} \\ \end{array}$$

(1)

$$\begin{array}{*{20}c} {H_{B} \left( f \right) = \mathop \sum \limits_{k = 0}^{N - 1} C_{k} e^{ - j2\pi k\tau f} } \\ \end{array}$$

(2)

Denoting the baseband EDSM signal and its spectrum by \(x_{B} \left( t \right)\) and \(X_{B} \left( f \right)\) and assuming all the mixer plus PA channels are identical with a large-signal gain *A*, the up-converted and combined RF signal at the output of the combiner is given by:

$$\begin{array}{*{20}c} {y\left( t \right) = e^{{j2\pi f_{c} t}} \mathop \sum \limits_{k = 0}^{N - 1} A C_{k} x_{B} \left( {t - k\tau } \right)} \\ \end{array}$$

(3)

Whose spectrum is given by:

$$\begin{array}{*{20}c} {Y\left( f \right) = AX_{B} \left( {f - f_{c} } \right)H_{B} \left( {f - f_{c} } \right)} \\ \end{array}$$

(4)

It is thus evident that the CANE filter transfer function \({ }H_{RF} \left( f \right)\) is an up-converted version of \(H_{B} \left( f \right)\) at the center frequency of \(f_{c}\), i.e.,

$$\begin{array}{*{20}c} {H_{RF} \left( f \right) = H_{B} \left( {f - f_{c} } \right)} \\ \end{array}$$

(5)

For instance, in a N-channel uniform combining CANE, let \(\tau\) denote the delay unit, the filter transfer function is

$$\begin{array}{*{20}c} {H_{RF} \left( f \right) = e^{{ - j\left( {N - 1} \right)\pi \left( {f - f_{c} } \right)\tau }} \frac{{\sin \left( {N\pi \left( {f - f_{c} } \right)\tau } \right)}}{{\sin \left( {\pi \left( {f - f_{c} } \right)\tau } \right)}} } \\ \end{array}$$

(6)

The rejection bands are thus located at:

$$\begin{array}{*{20}c} {f_{rej} = f_{c} \pm \frac{M}{N\tau }} \\ \end{array}$$

(7)

where \(M\) is an integer and \(M \ne kN\), \(k = 0,{ }1,{ }2,{ } \ldots\).

The first null-to-null bandwidth of the passband is

$$\begin{array}{*{20}c} {BW_{null} = \frac{2}{N\tau }} \\ \end{array}$$

(8)

From (7) and (8), all the rejection bands are symmetrically distributed with the center at \(f_{c}\), while a longer delay unit yields a narrower passband. It is also evident from (5) that the center frequency of the central pass band is solely determined by the LO frequency \(f_{c}\) and irrelevant to the choice of time delay. Therefore, one may change the filter bandwidth by tuning the delay length at baseband without disturbing the center frequency of the RF passband.

Figure 4 displays the simulated spectrum of CANE with EDSM. The testing signal is 1 MHz QPSK signal modulated by 2 level EDSM with a clock rate of 100 MSPS. The spectrum of the original EDSM signal without any noise suppression is plotted as the background in gray for comparison. Figure 4a displays the two-channel CANE with the second channel delayed by 5 clocks, i.e., 50 ns. It is observed that in 4(a), the first pair of nulls appear at \(f_{rej} = \pm 10MHz\) offsets from the center frequency, and thus, the first null to null bandwidth is 20 MHz. Figure 4b displays the spectrum with four-channel CANE combined with a uniform weighting at a delay increment of 30 ns. Compared with Fig. 4a and b, the four-channel case has provided better noise suppression over a wider frequency band.

To quantify the filtering performance of CANE, the signal-to-quantization noise ratio can be employed as a figure of merit. Denoting the complex, baseband spectrum of the bitstream modulated signal by \(S_{BMT} \left( f \right)\) and signal bandwidth by \(Sbw\), the in-band signal power \(P_{IBSIG}\) can be expressed as:

$$\begin{array}{*{20}c} {P_{IBSIG} = \mathop \smallint \limits_{ - Sbw}^{ + Sbw} \left| {S_{BMT} \left( f \right)} \right|^{2} df} \\ \end{array}$$

(9)

Note that CANE is often jointly used with an analogue output filter, it is important to evaluate the noise suppression within the analogue filter passband, as shown in Fig. 2. Let \(FBW\) be the bandwidth of the analogue filter, the in-band quantization noise power can thus be calculated by integrating the noise power within the analogue filter passband yet outside the desired signal band, i.e.,

$$\begin{array}{*{20}c} {P_{QNF} = \mathop \smallint \limits_{{ - \frac{FBW}{2} }}^{{ - \frac{Sbw}{2}}} \left| {S_{BMT} \left( f \right)} \right|^{2} df + \mathop \smallint \limits_{{ + \frac{Sbw}{2}}}^{{ + \frac{FBW}{2}}} \left| {S_{BMT} \left( f \right)} \right|^{2} df} \\ \end{array}$$

(10)

Therefore, the signal-to-quantization noise ratio subject to the analogue filter is defined as:

$$\begin{array}{*{20}c} {SQNR = \frac{{\mathop \smallint \nolimits_{ - Sbw}^{ + Sbw} \left| {S_{BMT} \left( f \right)} \right|^{2} df}}{{\mathop \smallint \nolimits_{{ - \frac{FBW}{2} }}^{{ - \frac{Sbw}{2}}} \left| {S_{BMT} \left( f \right)} \right|^{2} df + \mathop \smallint \nolimits_{{ + \frac{Sbw}{2}}}^{{ + \frac{FBW}{2}}} \left| {S_{BMT} \left( f \right)} \right|^{2} df}} } \\ \end{array}$$

(11)

For example, in the cases shown in Fig. 4a and b, the original EDSM signal has an unfiltered SQNR of 0.96 dB. With the two-channel CANE at 50 ns delay and 100 ns delay, the unfiltered SQNR is increased to 3.75 dB and 3.96 dB, respectively. In contrast, the original SQNR of EDSM signal within the 30 MHz analogue filter passband is 13.97 dB before CANE is applied. It has been improved to 20.33 dB and 17.9 dB, respectively, with the two-channel CANE at 50 ns and 100 ns delays. When the four-channel CANE with a delay increment of 30 ns is applied, the unfiltered SQNR becomes 7.43 dB. The filtered SQNR becomes 26.44 dB, which is about 12 dB improvement compared to the case without CANE.

It is also important to observe the CANE filter performance incorporating a physical BPF as a function of the delay length, which can help to find the best delay. Figure 5 plots the simulated SQNRs for different configurations with the 30 MHz BPF in place. From the plot, it is evident that CANE with more channels can offer better quantization noise suppression, as the four-channel CANE yields a higher SQNR compared to the two-channel case. The choice of delay may also be optimized for the best quantization noise suppression performance. A longer delay in CANE offers a narrower active filter passband with lower noise in this passband but a second or third digital passband may appear inside the analogue filter band, represented by the grating lobes of the SINC function in (6), while a shorter delay line may push those higher-order passbands outside the analogue filter band but at the price of an increased active filter passband with potentially higher residue noise in the band. In the two-channel case, 40nS delay on the second channel offers the best SQNR. Because the quantization noise from EDSM proportionally increases as frequency offset raises, the 40nS delay case places the digital filter stop band right next to the BPF cutoff frequency, where the noise is strongest.

Typically, higher in-band SQNR indicates that the PA delivers more power to the useful signal other than the noise and may lead to a better effective power efficiency. The in-band signal occupation \(\rho_{IBSO}\) can be used to show how much power filled into the desired signal band versus the total power output from the PA module.

$$\begin{array}{*{20}c} { \rho_{IBSO} = \frac{{\mathop \smallint \nolimits_{ - Sbw}^{ + Sbw} \left| {S_{BMT} \left( f \right)} \right|^{2} df}}{{\mathop \smallint \nolimits_{{ - \frac{FBW}{2} }}^{{ + \frac{FBW}{2}}} \left| {S_{BMT} \left( f \right)} \right|^{2} df}} \times 100\% } \\ \end{array}$$

(12)

The overall PA effective efficiency can be defined as the total power in the desired signal band divided by the DC power consumption, which can be equivalently estimated by the production of the PA efficiency and the in-band signal occupation.

### Digital control and calibration

As shown in Fig. 3, all the delay operations in a CANE transmitter can be implemented by digital signal processing without RF or analogue delay lines. Therefore, CANE filter can be reconfigured in a digital signal processor without physically changing the hardware. Digitally shifting registers that store the I and Q data in a DSP/FPGA processor is a relatively easy and time-efficient task. The resolution of the time delay setting can be as fine as one sampling clock. Since the baseband signal is identical for all the channels, the memory usage can be minimum. Figure 6 shows an example of such data storage/shifting architecture, where the I and Q data are shifted to the next register at each clock and the dth, the 2dth so on to the Ndth samples are output to the DACs corresponding to the multiple delayed channels with a time-delay interval of d clocks.

Since the delayed baseband signal is up-converted to RF, there might be phase error among the multiple LOs which will be carried over to the up-converted RF signals. In this system, the phase error generated by in each path mainly behaves as a constant angle offset. The phase error may affect the filtering performance and must be calibrated with a phase control circuitry at each LO path or simply with an arithmetic phase rotation between the I and Q paths at the input of each DAC by applying different weightings to these two data paths. For example, denote RF output signal with the phase and amplitude error in ith channel by

$$\begin{array}{*{20}c} {\widetilde{{y_{i} }}\left( t \right) = A_{e,i} y_{i} \left( t \right) \cdot e^{{j\phi_{e,i} }} } \\ \end{array}$$

(13)

where \(A_{e,i}\) represents the ratio of the non-calibrated amplitude for ith channel and the ideal one. \(A_{e,i} = 1\) if there is no amplitude error. \(\phi_{e,i}\) represents the phase offset of the ith channel. \(y_{i} \left( t \right)\) is the time domain signal output from ith PA path. To mitigate the phase and amplitude error, one can apply calibration on the baseband input before loading the signal to DAC, for ith channel, the corrected I and Q paths are given by:

$$\begin{array}{*{20}c} { \widetilde{{I_{i} }}\left( t \right) = \frac{1}{{A_{e,i} }}\left( {I_{i} \left( t \right)\cos \phi_{e,i} + Q_{i} \left( t \right)\sin \phi_{e,i} } \right)} \\ \end{array}$$

(14)

$$\begin{array}{*{20}c} {\widetilde{{Q_{i} }}\left( t \right) = \frac{1}{{A_{e,i} }}\left( {I_{i} \left( t \right){\text{sin }}\phi_{e,i} - Q_{i} \left( t \right)\cos \phi_{e,i} } \right)} \\ \end{array}$$

(15)

The calibration aligns the amplitude and phase of all the in-band signal after up-converting and amplification. The center frequency and the signals are combined in phase with theoretically no insertion loss. The digital calibration waives physical tuning of the RF circuitry, yet providing additional flexibility in RF signal control, such as multi-band EDSM signal generation and noise canceling [12], or fine tune of the center frequency of the passband.

### Power combiner design

After the RF signals with multiple delays are generated, a power combiner is used to combine the signals that are in-phase, i.e., in band, and reject those are out-of-phase, i.e., out of band. The suppression of out-of-band quantization noise can be accomplished in an absorptive fashion with a Wilkinson power combiner or in a reflective manner with a lossless power combiner. In a BMT, the power of quantization noise is a significant portion of that is being amplified. It must be reflected back to the power supply in order to maintain the high power efficiency. In the PLM PA [7], the recycling of the quantization noise power is achieved by utilizing a switching mode power amplifier with voltage source type output terminated by a current-rejection filter. The out-of-band signal at the PA thus sees a high impedance load which reflects its power back to the supply without causing dissipation. Similar strategies can be employed in CANE by properly designing the power combiner to exhibit a high impedance load to the out-of-phase or out-of-band signal, yet presenting the optimum load for in-phase or in-band signal.

For example, Fig. 7 shows a plot of the two PAs combined by a T-junction-based power combiner. For in-band signals, the two paths are delivering identical signals in a common mode while the out-of-band signals are combined anti-phase which follows a difference mode. The S-parameter of the two-way combiner is:

$$\begin{array}{*{20}c} {\left[ S \right] = \left[ {\begin{array}{*{20}c} 0 & { - \frac{j}{\sqrt 2 }} & { - \frac{j}{\sqrt 2 }} \\ { - \frac{j}{\sqrt 2 }} & \frac{1}{2} & { - \frac{1}{2}} \\ { - \frac{j}{\sqrt 2 }} & { - \frac{1}{2}} & \frac{1}{2} \\ \end{array} } \right]} \\ \end{array}$$

(16)

Port 1 is the output of the combiner, while port 2 and port 3 are connected to the output of two PAs. Assume the output is terminated by the optimum load and the two PA outputs have a phase difference due to delays, the incoming signal flowing into the three-port combiner network can be written as:

$$\begin{array}{*{20}c} {\left[ {V^{ + } } \right] = \left[ {0,V_{0} , V_{0} e^{ - j2\pi \tau \cdot \Delta f} } \right]^{T} } \\ \end{array}$$

(17)

where \(\tau\) is the baseband time delay on the second channel, and \(\Delta f\) refers to the frequency offset from the center. Note that the PLM PA is operating in the form of a switched voltage source, the incidence to the three-port combiner is thus represented by the voltages shown in Eq. (17). For the in-band signal, \(\Delta f\) is approximately zero; thus,

$$\begin{array}{*{20}c} {\left[ {V^{ + } } \right]_{inband} = \left[ {0, V_{0} , V_{0} } \right]^{T} } \\ \end{array}$$

(18)

The outgoing waves are given by:

$$\begin{array}{*{20}c} {\left[ {V^{ - } } \right]_{inband} = \left[ S \right]\left[ {V^{ + } } \right]_{inband} = \left[ { - j\sqrt 2 V_{0} , 0, 0} \right]^{T} } \\ \end{array}$$

(19)

This means that the PAs are terminated by the optimum matched load and their output is combined without loss.

For the out-of-band (OOB) signal, the outgoing wave is:

$$\begin{array}{*{20}c} {V_{1,OOB }^{ - } = - \frac{j}{\sqrt 2 }V_{0} \left( {1 + e^{ - j2\pi \tau \cdot \Delta f} } \right)} \\ \end{array}$$

(20)

$$\begin{array}{*{20}c} {V_{2,OOB }^{ - } = \frac{1}{2}V_{0} \left( {1 - e^{ - j2\pi \tau \cdot \Delta f} } \right)} \\ \end{array}$$

(21)

$$\begin{array}{*{20}c} {V_{3,OOB }^{ - } = \frac{1}{2}V_{0} \left( { - 1 + e^{ - j2\pi \tau \cdot \Delta f} } \right)} \\ \end{array}$$

(22)

Equation (20) reveals that when the two PAs are out of phase, the output power exhibits a periodically filtered pattern as shown in Fig. 8a.

In the rejection band, especially at the nulls of the filter, where the offset frequency satisfies the condition (7), two channels are out of phase by 180 degrees which forms a differential mode. An equivalent short circuit is formed at the combining junction which is later transferred to open circuit by the quarter wavelength transmission line looking out from the output of each PA. Denoting the input as \(\left[ {V^{ + } } \right]_{rej} = \left[ {0,V_{0} , - V_{0} } \right]^{T}\), the outgoing signals are obtained as,

$$\begin{array}{*{20}c} {\left[ {V^{ - } } \right]_{rej} = \left[ S \right]\left[ {V^{ + } } \right]_{rej} = \left[ {0, V_{0} , - V_{0} } \right]} \\ \end{array}$$

(23)

This proves that the two input ports now exhibit open-circuit impedance for the differential mode. The input impedance of the combiner versus frequency is derived by simulating the phase and magnitude of the reflection coefficient at one of the input ports, which is plotted against the filter transfer function in Fig. 8b. At the center frequency, the filter transfer function has a magnitude about 1 that indicates all the signal power is delivered to the load. In the filter rejection band, the equivalent reflection coefficient at port 2 and port 3 is:

$$\begin{array}{*{20}c} {\Gamma_{2} = \Gamma_{3} = 1} \\ \end{array}$$

(24)

This indicates that the output of each PA at this frequency is indeed completely reflected by the open circuit. The same principle can be utilized to design power combiners for more channels. Figure 9 shows a design of four-channel power combiner. One may prove that the load impedance of Z_{opt} is transformed to Z_{opt} at each of the 4 input ports (reference plane A) in common mode. The input impedance is open circuit for difference modes between any of the two branches, as short circuit is formed either in reference plane B or plane C.

### Bandwidth consideration

The bitstream modulated transmitter includes baseband sampling, RF amplification and filtering. Each part in the signal path may affect the system bandwidth. In this work, the RF components are designed to offer sufficient bandwidth. The PLM PA units operate from 1.8 to 2.1 GHz. The four-way combiner including multi-section transmission line-based impedance transformers is with a wide bandwidth as shown in Fig. 10. The bandwidth limitation relevant to the proposed transmitter is twofold. The first is the limited bandwidth of the bitstream modulation generator. The clock rate of EDSM needs to be much higher (8 to 10 times greater) than the envelop bandwidth to shape the quantization noise away from the signal band. It is also desired that the sampled waveform to be rectangular pulse trains with fast rising and falling edges. This further elevates the desired sampling rate in a digital implementation. The second bandwidth limitation is limited by the sampling rate of CANE processor which is limited by the technology of the controller implementation (DSP, FPGA or ASIC). The maximum sampling rate of the current FPGA CANE processor is 100MSPS which limits the signal bandwidth to be less than 50 MHz.

In the current setup, the EDSM generator bandwidth limit is dominant. While various envelope modulation techniques [8,22,-24] or a mixed signal EDSM generator implementation could be used to further reduce the noise level and the required sampling rate, we choose 1.2 MHz QPSK signal with 5.3 dB PAPR and 1.4 MHz LTE signal with 10 dB PAPR to demonstrate the proposed concept. Future work will consider higher sampling rate platform and different envelope modulation technique.